�������� Mathematics for Computer Science Well Ordering principle MIT 6.042J/18.062J Every nonempty set of nonnegative integers The Well Ordering has a Principle least element. Familiar? Now you mention it, Yes. Obvious? Yes. Trivial? Yes. But watch out: This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License . Albert R Meyer February 13, 2012 Albert R Meyer February 13, 2012 Lec 2M.1 Lec 2M.2 Well Ordering principle Well Ordering principle Every nonempty set of Every nonempty set of p y nonnegative integers nonnegative integers integers rationals has a has a least element. least element. NO! NO! Albert R Meyer February 13, 2012 Albert R Meyer February 13, 2012 Lec 2M.3 Lec 2M.4 What is the N ::= nonnegative integers • youngest age of MIT For rest of this talk, graduate? • smallest # neurons in “number” means any animal? nonnegative integer • smallest #coins = $1.17? Albert R Meyer February 13, 2012 Albert R Meyer February 13, 2012 Lec 2M.5 Lec 2M.6 ��
�������� 2 proof used Well Ordering Proof using Well Ordering m Find smallest number m s.t. Proof : …suppose 2 = n m . If m, n had a 2 = …can always find such m, n > 0 n without common factors… common factor, c > 1, then why always ? ( ) m / c and m/c m 2 = < ( ) n / c Albert R Meyer February 13, 2012 Albert R Meyer February 13, 2012 Lec 2M.7 Lec 2M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 M 7 7 Lec 2M Lec 2M.8 M 8 M 8 M 8 M 8 M 8 M 8 8 8 Proof using Well Ordering Find smallest number m s.t. m 2 = . n This contradiction implies m, n have no common factors. Albert R Meyer February 13, 2012 Lec 2M.9 Lec 2M 9 M 9 M 9 M 9 M 9 M M 9 M 9 M 9 M 9 9 9 9 ��
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